Add-ons: compactness measured by s-numbers #
Auxiliary Hilbert/Banach-space material around the s-numbers framework, following Pietsch, Eigenvalues and s-numbers (Cambridge, 1987), §2.11.
Layout #
AddOns.Approximable— the class of approximable operators (those whose approximation numbersaₙtend to zero), its closure properties, the implicationIsApproximable ⇒ IsCompactOperator, and the special caseisCompactOperator_of_rank_le(finite rank ⇒ compact) used by the exampleSNumbers.Examples.IdentityL1Linfty.AddOns.Compact— which s-number sequences detect compactness. On every Banach spaceSis compact iffcₙ(S) → 0, iffdₙ(S) → 0. For the approximation numbers this fails (Enflo), but on Hilbert spaces every s-number sequence works, since all of them agree withaₙthere.
The singular value decomposition itself — including the scalar
factorisation SVD.exists_scalar_factorisation consumed by the
s-numbers uniqueness theorem — lives in BasicResults.SVD.